Expressions

Lcapy expressions are similar to SymPy expressions except they have a specific domain depending on the predefined variables t, s, f, omega, and jomega.

Symbols

Lcapy has a number of pre-defined constants, variables, and functions.

Constants

  • pi 3.141592653589793...
  • j \(\sqrt{-1}\)
  • oo infinity
  • zoo complex infinity

Variables

Lcapy has five predefined variables:

  • s Laplace domain complex frequency
  • f Fourier domain frequency
  • t time
  • omega Fourier domain angular frequency
  • jomega Fourier domain angular frequency times j

A time-domain expression is produced using the t variable, for example,

>>> v = exp(-3 * t) * u(t)

Similarly, a Laplace-domain expression is produced using the s variable, for example,

>>> V = s / (s**2 + 2 * s + 3)

Symbols can also be created with Lcapy’s symbol function:

>>> tau = symbol('tau', real=True)

They are also implicitly created using Lcapy’s expr function:

>>> v = expr('exp(-t / tau) * u(t)')

Note, symbols created with symbol and expr are assumed to be positive, unless explicitly specified not be.

Mathematical functions

Lcapy has the following built-in functions: sin, cos, tan, atan, atan2, gcd, exp, sqrt, log, log10, Heaviside, H, u, DiracDelta, delta, and conjugate.

Printing functions

  • pprint pretty print an expression
  • latex convert an expression to LaTeX string representation
  • pretty convert an expression to a string with a prettified form

Utility functions

  • symbol create a symbol
  • expr create an expression

Note, sympy does not allow symbol names that are Python keywords. For example, expr(‘is(t)’) fails. A workaround is to use an underscore in the name, for example, expr(‘i_s(t)’).

Transformation and substitution

Substitution and transformation use a similar syntax V(arg). If arg is t, f, s, omega, or jomega, transformation is performed, otherwise substitution is performed. This behaviour can be explicitly controlled using the subs and transform methods, for example,

>>> from lcapy import *
>>> V1 = Vsuper('3 * exp(-2 * t)')
>>> V1.transform(s)
  3
─────
s + 2
>>> V1.transform(t)
   -2⋅t
3⋅e
>>> V1.subs(2)
   -4
3⋅e

Transformation

  • \(V(t)\) returns the time domain transformation
  • \(V(f)\) returns the Fourier domain transformation
  • \(V(s)\) returns the Laplace domain (s-domain) transformation
  • \(V(omega)\) returns the angular Fourier domain transformation
  • \(V(jomega)\) returns the angular Fourier domain transformation obtained from the Laplace domain transformation with \(s = j \omega\).

For example:

>>> from lcapy import *
>>> V1 = Vsuper('3 * exp(-2 * t)')
>>> V1(t)
   -2⋅t
3⋅e
>>> V1(s)
  3
─────
s + 2

Substitution

Substitution replaces sub-expressions with new sub-expressions in an expression. It is most commonly used to replace the underlying variable with a constant, for example,

>>> a = 3 * s
>>> b = a(2)
>>> b
6

Evaluation

Evaluation is similar to substitution but requires all symbols in an expression to be substituted with values. The result is a numerical answer. The evaluation method is useful for plotting results. For example,

>>> a = expr('t**2 + 2 * t + 1')
>>> a.evaluate(0)
1.0

The argument to evaluate can be a scalar, a tuple, a list, or a NumPy array. For example,

>>> a = expr('t**2 + 2 * t + 1')
>>> tv = np.linspace(0, 1, 5)
>>> a.evaluate(tv)
array([1.    , 1.5625, 2.25  , 3.0625, 4.    ])

Phasors

Phasors represent signals of the form \(v(t) = A \cos(\omega t + \phi)\) as a complex amplitude \(X = A \exp(\mathrm{j} \phi)\) where \(A\) is the amplitude, \(\phi\) is the phase, and the angular frequency, \(\omega\) is implied.

The signal \(v(t) = A \sin(\omega t)\) has a phase \(\phi=-\pi/2\).

Assumptions

SymPy relies on assumptions to help simplify expressions. In addition, Lcapy requires assumptions to help determine inverse Laplace transforms.

There are several attributes for determining assumptions:

  • is_dc – constant
  • is_ac – sinusoidal
  • is_causal – zero for \(t < 0\)
  • is_real – real
  • is_complex – complex
  • is_positive – positive
  • is_integer – integer

For example:

>>> t.is_complex
False
>>> s.is_complex
True

Assumptions for symbols

The more specific assumptions are, the easier it is for SymPy to solve an expression. For example,

>>> C_1 = symbol('C_1', positive=True)

is more appropriate for a capacitor value than

>>> C_1 = symbol('C_1', complex=True)

Notes:

  1. By default, the symbol and expr functions assume positive=True unless real=True or positive=False are specified.
  2. SymPy considers variables of the same name but different assumptions to be different. This can cause much confusion since the variables look identical when printed. To avoid this problem, Lcapy creates a symbol cache for each circuit. The assumptions associated with the symbol are from when it is created.

The list of explicit assumptions for an expression can be found from the assumptions attribute. For example,

>>> a = 2 * t + 3
>>> a.assumptions
{'real': True}

The assumptions0 attribute shows all the assumptions assumed by SymPy.

Assumptions for inverse Laplace transform

The unilateral Laplace transform ignores the function for \(t < 0\). The unilateral inverse Laplace transform thus cannot determine the result for \(t <0\) unless it has additional information. This is provided using assumptions:

  • causal this says the signal is zero for \(t < 0\).
  • ac this says the signal is sinusoidal.
  • dc this says the signal is constant.

For example,

>>> H = 1 / (s + 2)
>>> H(t)
⎧ -2⋅t
⎨e      for t ≥ 0

>>> H(t, causal=True)
 -2⋅t
e    ⋅Heaviside(t)
>>> h = cos(6 * pi * t)
>>> H = h(s)
>>> H
    s
──────────
 2       2
s  + 36⋅π
>>> H(t)
{cos(6⋅π⋅t)  for t ≥ 0
>>> H(t, ac=True)
cos(6⋅π⋅t)

Classes

Lcapy uses myriads of classes, one for each combination of domain (time, Fourier, Laplace, etc) and expression type (voltage, current, impedance, admittance, transfer function). For example, to represent Laplace domain entities there are the following classes:

  • sExpr generic Laplace-domain expression
  • Vs Laplace-domain voltage
  • Is Laplace-domain current
  • Hs Laplace-domain transfer function
  • Ys Laplace-domain admittance
  • Zs Laplace-domain impedance

SymPy

The underlying SymPy expression can be obtained using the expr attribute of an Lcapy expression. For example,

>>> a = 2 * t + 3
>>> a.expr
2⋅t + 3

The methods of the SymPy expression can be accessed from the Lcapy expression, for example,

>>> a.as_ordered_terms()
[2⋅t, 3]

Another example is accessing the assumptions that SymPy considers:

>>> t.assumptions0
{'commutative': True,
 'complex': True,
 'hermitian': True,
 'imaginary': False,
 'real': True}

Note, every real symbol is also considered complex although with no imaginary part. The proper way to test assumptions is to use the attributes is_complex, is_real, etc. For example,

>>>t.is_real True >>>t.is_complex False